Tensor calculus

In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his theory of general relativity. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.

Tensor calculus has many real-life applications in physics and engineering, including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), and general relativity (see mathematics of general relativity).

See also

• Vector analysis
• Matrix calculus
• Ricci calculus
• Tensors in curvilinear coordinates
• Multilinear subspace learning

Further reading

• Dullemond, Kees; Peeters, Kasper (1991–2010). "Introduction to Tensor Calculus" (PDF). Retrieved 17 May 2018.CS1 maint: Date format (link)

Books

• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
• Sokolnikoff, Ivan S (1964). Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. John Wiley & Sons Inc; 2nd edition. ISBN 0471810525.
• J.R. Tyldesley (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
• D.C. Kay (1988). Tensor Calculus. Schaum’s Outlines, McGraw Hill (USA). ISBN 0-07-033484-6.
• P.Grinfeld (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 1-4614-7866-9.